Diversity systems are used for fading compensation (See, D. G. Brennan, "Linear diversity combining techniques", Proc. of IRE, Vol. 47, pp. 1075-1102, June 1959.). In a dual diversity system, one symbol is transmitted over two channels and respective signals are combined at a receiver. There are several types of diversity combining techniques in practical use: selection combining (SC), equal gain combining (EGC), and maximal ratio combining (MRC). In the SC technique, a channel with the largest signal to noise ratio is selected. SC is simply implemented with orthogonal signaling and noncoherent demodulation which are frequently used in fading channels (G. Chyi, J. G. Proakis, and C. M. Keller, "On the symbol error probability of maximum selection diversity reception schemes over a Rayleigh fading channel", IEEE Trans. Commun. Vol. 37, No. 1, pp. 78-83, January 1989.). The most efficient communication system design for M-ary orthogonal channels with noncoherent demodulation would employ low rate codes over a Galois field GF(q) with M=q (W. E. Ryan and S. G. Wilson, "Two classes convolutional codes over GF(q) for q-ary orthogonal signaling", IEEE Trans. Commun. Vol. 39, No. 1, pp 30-40, January 1991.).
There are two kinds of coding techniques for dual diversity systems: one has each code symbol transmitted twice over each channel and the other has transmitted symbols that are made to differ from channel to channel through a special channel coding operation (G. Benelli, "Two coding techniques for diversity communications systems", IEEE Trans. Commun. Vol. 38, No. 9, pp. 1530-1538, September 1990.). In the second technique, it is generally required to use multi-level coding operations.
When a codeword c is transmitted over a communication channel, channel noise may corrupt the transmitted signals. As a result, a receiver receives the corrupted version of the transmitted codeword c+e, where e is an error pattern of some weight u. The result of bounded distance decoding is categorized into three types: correct decoding, decoding failure, and decoding error. If u.ltoreq.t, then a bounded distance decoder on the receiver'send detects and corrects the error e and recovers c. If u&gt;t, then the decoder fails and either detects the presence of the error pattern but is unable to correct it, or miscorrects the received pattern c+e for some other codeword c.sup.l +L if the received pattern falls into the radius l Hamming sphere of c.sup.l +L . Miscorrection is more serious than error detection. This can occur when an error pattern e is of weight u&gt;t+1. Assuming that all error patterns of weight u are equally probable, decoding failure occurs when a received word is not contained in a decoding sphere of any codeword. Decoding failure is a kind of error detection. The conditional probability of decoding failure and undetected error are related with the geometry of code in a vector space GF(q).sup.n. Let P.sub.E (u) denote the decoder error probability given that an error pattern of weight u occurs. P.sub.E (u) is given by the ratio of the number of decodable words of weight u to the number of words of weight u in the whole vector space. Let Q be the probability that a completely random error pattern will cause a decoder error. Then Q is the ratio of the number of decodable words to the cardinality of the whole vector. It is given by ##EQU1##
It is known that P.sub.E (u) is less than or approximately equal to Q which is independent of u for Reed-Solomom codes (K. M. Cheung, "More on the decoder error probability for Reed-Solomon codes", IEEE Trans. Inform Theory, Vol. 35, No. 4, pp. 895-900, July 1989.).